Everybody was puzzle fighting
Easily, the true star of the Capcom Puzzle World show is Super Puzzle Fighter II Turbo. It comes in three flavors: X, Y and (wait for it ) Z. These three different modes play surprisingly differently than each other, giving Puzzle Fighter the most bang for your buck. In the X mode, your goal is to create fatty gems by dropping similarly colored gems into blocks (2x2, 2x3, 4x8, etc.) and then detonating them by dropping special blocks on top of your gem constructions. Every time you shatter your gems with a special block, your opponent gets a shower of temporarily unbreakable blocks that screw up their own gem constructs (and they can mess you up in a similar fashion whenever they shatter their own gems).
The Y mode plays similar to the Sega Genesis classic Columns where you goal is to simply get three gems lined up horizontally, vertically or diagonally. Once the gems are lined up, they shatter and the cascade of unbreakable blocks falls onto your hapless opponent.
The Z mode seems least like the other modes, because the gems slowly crawl up from the bottom. In order to stem the rising tide of pesky jewels, you can rotate four blocks either left or right to match up colors and create larger gem constructs. Again, your goal is to shatter your gems with special blocks before your opponent does, but this mode requires that you do it by shifting the gems around in a manner similar to Hexic.
While Puzzle Fighter contains its three modes under one puzzle fighting umbrella, Buster Bros., Super Buster Bros. and Buster Buddies take a different approach and try to break one game into three. All three of these games are basically the same: you have a little dude that shoots tethered harpoons straight up into the air so that he can bust lethal balloons into their smaller constituent balloons before they bounce on him and kill him. The key difference between the three versions is that they have different titles and different background images for each stage. Of the three, Buster Buddies looks like the most interesting of the bunch because it features famous works of fine art as its background images (and also because it's the only Buster game with a tutorial).